Nuprl Lemma : continuous-minus
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ⇒ -(f[x]) continuous for x ∈ I)
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rminus: -(x),
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
all: ∀x:A. B[x],
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
label: ...$L... t,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
radd_comm,
rminus-rminus,
req_transitivity,
req_inversion,
rminus-as-rmul,
rmul_functionality,
radd_functionality,
req_weakening,
rminus-radd,
rabs_functionality,
rleq_functionality,
uiff_transitivity,
rmul_wf,
radd_wf,
iff_weakening_equal,
rabs-rminus,
true_wf,
squash_wf,
interval_wf,
rfun_wf,
continuous_wf,
icompact_wf,
set_wf,
nat_plus_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
i-member-approx,
rminus_wf,
less_than_wf,
all_wf,
int-to-real_wf,
rless_wf,
real_wf,
i-approx_wf,
i-member_wf,
rsub_wf,
rabs_wf,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberFormation,
productElimination,
independent_pairFormation,
because_Cache,
independent_functionElimination,
lemma_by_obid,
isectElimination,
productEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_set_memberEquality,
applyEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
minusEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. (f[x] continuous for x \mmember{} I {}\mRightarrow{} -(f[x]) continuous for x \mmember{} I)
Date html generated:
2016_05_18-AM-09_11_38
Last ObjectModification:
2016_01_17-AM-02_38_34
Theory : reals
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